Simple CAD model for direct coupled doublesplit ring resonators

A.D. Alwakil and A.M.E. Safwat

An analytical coupled-line based model is proposed for arbitrarydouble split ring resonator configurations. The model is verified bydesigning a slow-wave based Butterworth lowpass filter. The filterhas been fabricated and measured. The measurements are in a goodagreement with theoretical results over a wide frequency range.

Introduction: Analytical models for planar split ring resonators (SRRs)have been widely investigated [1–5]; however, they cannot provide,simultaneously, simple and intuitive physical significance. Forexample, lumped-component based models [1, 5] are limited by the res-onance frequency prediction and do not include higher-order dynamicresonances. Differential-equation based models [2] suffer from com-plexity for complicated SRR shapes, while coupled-line based models[3, 4] are limited to microstrip-like SRRs only.

In this Letter, we propose a new analytical coupled-line based modelfor directly coupled double SRRs with different geometrical configur-ations. An SRR is first modelled as coupled line sections, the equivalentT-network is derived, and then the limiting condition of ground removalis applied to extract the equivalent circuit. This model can be alsoextended for indirect coupling cases as well.

Proposed model: Considering lossless four terminal symmetric coupledlines, the even and odd modes have, respectively, Ze, Zo characteristicimpedances and be, bo propagation constants. These parameters arewritten in terms of Lo and Co, the inductance and capacitance per unitlength of an isolated single line, and Lm and Cm, the mutual inductanceand capacitance per unit length, as follows [6]:

Ze =����������L0 + Lm

C0 − Cm

√(1)

Ze =����������L0 − Lm

C0 + Cm

√(2)

be = v����������������������(L0 + Lm)(C0 − Cm)

√(3)

be = v����������������������(L0 − Lm)(C0 + Cm)

√(4)

By removing the ground or in terms of circuit parameters removing thecapacitance per unit length of the even mode, i.e. (Co–Cm) tends to zero;at this limit, Ze tends to infinity and be tends to zero, with the productZebe being the inductive term. So, depending on the split ring configur-ation and the excitation terminals (same side, same line or diagonal) [4],the even mode inductive effect would exist or vanish, which has animpact on the input impedance at the excitation terminals.

Case study: To clarify this concept, the model is applied to the doublering case where the excitation terminals are as shown in Fig. 1a. Aftersome mathematical manipulations the equivalent T-network, [7], isgiven by:

Z1 = jZ0 tan u0 (5)

Z2 = j

2Ze tan ue −

j

2Z0 tan u0 (6)

where ue and uo are the even and odd mode electrical lengths, respect-ively. Applying the limits Ze � 1, ue � 0 leads to:

limZe�1ue�1

Ze tan ue = limx�0

j

���������L0 + Lm

x

√tan v

l

2

������������(L0 + Lm)x

√( )

= jv(L0 + Lm)l

2

(7)

Zin = j2v(L0 + Lm)l//j4Z0 tan u0 (8)

where x ¼ Co–Cm and Zin is the input impedance. So the equivalentcircuit is a lumped inductor in parallel with a short-circuited stub oflength equal half the average length of the double SRR as shown inFig. 1b. This procedure can be extended to different SRR configurations.Table 1 shows the equivalent circuit models of some of them.

ELECTRONICS LETTERS 10th May 2012 Vol. 48

1

1

a b

2

2

2(L0 + Lm)Δl

2

4Z0

Δl

Fig. 1 Double split ring resonator schematic and proposed equivalentcircuit model

a Double split ring resonator schematicb Proposed equivalent circuit model

Table 1: Equivalent circuit model of different SRRs using proposedmodel assuming that gap and parasitic capacitances areignored

Number Cell schematic Equivalent circuit

1 1

1

1

1

2

44

4 4

2

2

2

2

2

2

12

12

2

3

Δl

Δl

ΔlΔl

Δl

Δl

Δl

2Δl

Z0

2Z0

2Z0

2Z0

2Z0

(L0 + Lm)

(L0 + Lm)

(L0 + Lm)

Lowpass filter design: To prove the efficiency of the proposed model, aslow-wave based lowpass filter, where a host transmission line is loadedperiodically with series inductances, is designed. Bloch impedance andcutoff frequency are chosen to be 50 V and 2.2 GHz, respectively. Theprototype of the unit cell is shown in Fig. 2a where a host line with28.5 V characteristic impedance is loaded by L ¼ 4.77 nH. To realisethis unit cell on RT/duroid 6010 substrate (h ¼ 0.64 mm, er¼ 10.2and 0.023 tangent loss), the microstrip host line has 1.6 mm widthand the unit cell pitch size ( p) is chosen to be 10.5 mm. The inductorconsists of a double SRR with a window in the ground, the width andseparation of the lines are 0.5 mm and 0.3 mm, respectively, to haveZo, Ko and Lo + Lm equal to 40 V, 5.85 and 3.17 nH where Ko is theodd mode effective dielectric constant. The average length of the ringis chosen to be 10.34 mm to get the cutoff frequency at 2.2 GHz. Allother dimensions are shown in Fig. 2b.

8mm

8mm

1mm

1mm

0.3mm

width=0.5mmseparation=0.3mm

mean radius=1.6mm

L=4.77nH Ze=67.8Z0=40Ke=7.3Ko=5.85

p

Zc=28.5

p=10.5mm

host linewidth=1.6mm

a b

Fig. 2 Unit cell prototype, and layout of unit cell using double SRR

a Unit cell prototypeb Layout of unit cell using double SRR

Three cells are cascaded to constitute the lowpass filter. Fig. 3a showsthe scattering parameters of the equivalent circuit and EM simulationsusing HFSS (ver. 13) simulator. A small deviation between EM andcircuit simulations, which is due to radiation, is observed around5 GHz. Otherwise the agreement is very good. The filter is fabricatedand measured. Fig. 3b shows the measured data.

Measurements deviate from the EM simulations shown in Fig. 3a. Bychecking the datasheet of the RT/duroid 6010, it was stated that the per-mittivity may vary from the nominal value (10.2) to 10.9. The filter wasresimulated with the new value of the substrate permittivity. EM

No. 10

simulation results are shown in Fig. 3b and they are in a good agreementwith measurements.

40

20

0

–20

200

–20

–40

–60

–80

10

–10

–20

–30

–40

0

–400 1 2 3 4 5

S11

, dB

S11

, dB

S12

, dB

0

–20

–30

–60

–50

–40

S12

, dB

frequency, GHz

0 1 2 3 4

frequency, GHz

cricuit modelEM

meas.EM

a b

Fig. 3 S-parameters of EM and circuit simulations of lowpass filter(er ¼ 10.2) and of EM simulations and measurements of three cascadedlow pass filter (er ¼ 10.9)

a S-parameters of EM and circuit simulations of lowpass filter (er ¼ 10.2)b S-parameters of EM simulations and measurements of three cascaded lowpassfilter (er ¼ 10.9)

Conclusion: Proposed is a geometrical circuit model for direct coupledsplit ring resonators. It is based on coupled-lines where the split ringresonator was considered as a limiting case for its microstrip versionafter removing the ground. A double split ring resonator was investi-gated as a case study and its equivalent circuit was derived. Based onthis model, a slow-wave Butterworth three pole lowpass filter hasbeen designed. Simulation and experimental results are provided andare in agreement with theoretical expectations.

Acknowledgment: This work was supported by the Science andTechnology Development Fund (STDF), Cairo, Egypt.

ELECTRO

# The Institution of Engineering and Technology 201218 March 2012doi: 10.1049/el.2012.0846One or more of the Figures in this Letter are available in colour online.

A.D. Alwakil and A.M.E. Safwat (Electronics and CommunicationEngineering Department, Faculty of Engineering, Ain ShamsUniversity, 1 El-Sarayat St, Abbassia, Cairo 11517, Egypt)

E-mail: asafwat@ieee.org

References

1 Baena, J.D., Bonache, J., Martin, F., Sillero, R.M., Falcone, F., Lopetegi,T., Laso, M.A.G., Garcia-Garcia, J., Gil, I., Portillo, M.F., and Sorolla,M.: ‘Equivalent-circuit models for split-ring resonators andcomplementary split-ring resonators coupled to planar transmissionlines’, IEEE Trans. Microw. Theory Tech., 2005, 53, (4), pp. 1451–1461

2 Shamonin, M., Shamonina, E., Kalinin, V., and Solymar, L.: ‘Resonantfrequencies of a split ring resonator: analytical solutions and numericalsimulations’, Microw. Opt. Technol. Lett., 2005, 44, (1), pp. 133–136

3 Zhurbenko, V., Jensen, T., Krozer, V., and, and Meincke, P.: ‘Analyticalmodel for double split ring resonators with arbitrary ring width’, Microw.Opt. Technol. Lett., 2008, 50, (2), pp. 511–515

4 Safwat, A.M.E., and Abuelelfadl, T.M.: ‘Coupled lines from filter tocomposite right/left handed cell’, PIER B, 2010, 26, pp. 451–469

5 Martel, J., Marques, R., Falcone, F., J.D., Baena, J.D., Medina, F.,Martin, F., and Sorolla, M.: ‘A new LC series element for compactbandpass filter design’, IEEE Trans. Microw. Theory Tech., 2004, 14,(5), pp. 210–212

6 Paul, C.R.: ‘Analysis of multiconductor transmission lines’ (Wiley,New York, USA, 1994)

7 Zysman, G.I., and Johnson, A.K.: ‘Coupled transmission line networks inan inhomogeneous dielectric medium’, IEEE Trans. Microw. TheoryTech., 1969, 17, (4), pp. 753–759

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